Harmonic function

Definition

On an open subset in the complex numbers

Let $U$ be an open subset in $\mathbb {C}$. A $C^{2}$-function $f:U\to \mathbb {R}$ is termed a harmonic function if we have:

$\Delta f:={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}=0$

where equality holds identically, at all points of $U$.

On an open subset in a real vector space

Let $U$ be an open subset of $\mathbb {R} ^{n}$, and $f:U\to \mathbb {R}$ be a $C^{2}$-function (i.e. $f$ is twice continuously differentiable). Then, we say that $f$ is harmonic if we have:

$\Delta f:=\sum _{k=1}^{n}{\frac {\partial ^{2}f}{\partial x_{k}^{2}}}=0$

where equality must hold at all points of $U$.